Mathematics stands out for its universality. Did Galileo not already say that "the book of Nature is written in the language of mathematics"? From its invention onward, mathematics has been used for trade, for the construction of the pyramids and the International Space Station, and for vehicles on land, sea and air. In the 20th century, alongside the rise of computing, operations research aimed at improving systems underwent explosive growth, as did statistical data processing.
Creating value --------------------
The 2019–2020 Year of Mathematics is certainly a good time to discover the sometimes overlooked uses of mathematics\* in telecommunications, artificial intelligence (AI), meteorology, the study of how planets and stars form, the optimization of factory operations or medical treatments, epidemic management, imaging, tsunami warnings, smart cities…
The importance of mathematics in digital technology (see FOCUS) and finance was already well known. Recent special issues of Tangente have highlighted its use in medicine, economics, and the climate and energy transition. Mathematics likewise provides a genuine competitive advantage throughout today’s economy: industrial sciences, physics, biology, chemistry… It has enabled new markets and professions to emerge, with AI and data science among the latest examples. Mathematics is involved in innovation at every level of a company’s value chain, whatever its sector. In product design, it helps reduce costs and accelerate research. In production, mathematical models of parts and assembly processes help optimize manufacturing methods. In quality control, statistics and image-analysis technologies allow production to be monitored remotely, thereby improving its efficiency. Management uses mathematics to keep energy costs under control and improve financial oversight. Marketing and sales departments value it for effectively modeling customer relationships and analyzing customer behavior, as well as for pricing management. Strategy and foresight departments use it to improve risk management and decision-support processes.
From multinationals to SMEs -----------------------------
Today, companies operate within increasingly complex ecosystems, whatever their size. This trend is likely to intensify in the years ahead. Mathematics provides unrivaled tools for analyzing and managing this environment: signal and image processing; data mining; AI; modeling, simulation and optimization; high-performance computing; statistics and stochastic calculus; information security and cryptography… Large corporations and start-ups alike have long relied on mathematics to boost innovation. Now, mathematics offers solutions at every level of a company’s value chain, from developing new products through manufacturing processes to bringing those products to market. Companies of every size can therefore benefit from mathematical expertise. SMEs in particular operate in increasingly complex markets, and the analytical, simulation and forecasting tools provided by mathematics offer them undeniable competitive advantages.
Which educational pathways? -------------------------------
So how can students enter these professions? After completing a science-focused baccalauréat, students can enroll in a university bachelor’s degree in mathematics, followed by a specialized master’s degree in scientific computing, cryptography, or statistics and data processing. Graduates of other science programs, such as physics or mechanics, can also enter a multidisciplinary master’s program in scientific computing. Similarly, graduates in economics and the social sciences can enter multidisciplinary master’s programs in statistics. Some engineering schools also offer an applied mathematics track following a classe préparatoire (post-secondary preparatory program), whether integrated into the school or completed elsewhere.
Several engineering schools offer arrangements allowing students to complete part or all of a master’s degree in applied mathematics, sometimes leading to a dual degree. Some of these programs combine work and study, while others can be completed remotely. These five-year programs qualify graduates to apply for company positions as mathematicians in engineering-level roles in research and development (R&D), scientific computing, data analysis, statistics or cryptography. This five-year course of study can be followed by a doctorate, which generally takes three years; it provides training through research under a fixed-term contract. The doctorate can be completed in partnership with a company through the CIFRE scheme (industrial agreement for research training), which subsequently makes it easier to secure a job in industry… though not necessarily at a higher-education institution.
Those wishing to supplement their initial education by acquiring mathematical expertise can turn to continuing education programs offered by higher-education institutions. Many master’s programs are directly accessible in this way. These demanding programs may require a substantial time commitment that is not always compatible with working at the same time, unless arrangements can be made to spread the program over a longer period or the student can take special leave. Another option is to take short, employer-funded courses lasting a few days. These can provide the foundations needed to enter a new field or enable participants to specialize in a cutting-edge technique.
\* For a sense of the scale, see the following brochures: L’explosion des mathématiques (2002): **smai.emath.fr/spip/documents/explosion\_math.pdf** Mathématiques, l’explosion continue (2013): smai.emath.fr/documents/math\_explosion2013.pdf
*Nicolas Nguyen is president of the Le Temps des Sciences association (see "Days devoted to innovation"). Fabrice Mahé is co-director of Agence Lebesgue, mathematics for innovation.*